Download Chapter 4, Section 6

Math 1330, Chapter 2, Section 2
All of the following have something in common
y5
y  3x  2
y  2x 2  4x  10
y  x 7  11
All are functions that are POLYNOMIALS…there are real number coefficients and the
exponents of x are whole numbers (in the text, “nonnegative integers”). When you graph
these you start somewhere and draw a smooth line with no loops or back tracking
(remember each is a function) and your pencil never leaves the paper while
sketching…it’s one continuous line that you draw…and there are no sharp corners (called
“cusps”).
The first function has degree zero (horizontal line)…next to it is a function of degree 1 (a
line with a non zero slope)…next is a parabola, aka quadratic, aka polynomial of degree
2, and last is a polynomial of degree 7. The highest power of x in the summands is
written first in line and is called the “degree” of the polynomial. The term with the
highest power is called the “leading term”. The number multiplier in the leading term is
called the “leading coefficient”.
I want you to become comfortable graphing polynomials without calculating any points
at all.
The first thing to learn is about the behavior of the graph at the extreme x values.
If the degree of the polynomial is even (2, 4, 6, ….) then
Both endpoints point up if the leading coefficient is positive
See x 2
Both endpoint point down if the leading coefficient is negative
See  x 2
1
If the degree of the polynomial is odd (1,3, 5, …) then
If the leading coefficient is positive it’s left down and right up
See x 3
If the leading coefficient is negative, it’s left up and right down
See  x 3
Graphs that are “like x 2 ” are graphs that have higher even numbers as exponents and no
middle terms…these are flatter at the turn around point than x 2 , though. The same holds
true for x 3 and higher odd powers.
Check out
f ( x )  ( x  2) 6
And f (x)  (x  3) 7
End behavior is:
End behavior is:
2
Next we’ll be able to tell where the graph is and is not once we get the graph in factored
form:
Let’s look at f ( x )  ( x  1)( x  2)( x  5)
What’s the leading term?
What’s the end behavior?
What’s the last term?
is it a constant? is it times a power of x?
What’s the y intercept?
What are the x intercepts? aka “critical points”
Put them on an axis and let’s do some analysis
Pick four test points around the “crits” and figure out where the curves of the polynomial
go using these TP to get the sign of the y value (+ above x axis,  below x axis)
3
TP one < 5
What is the excluded area?
5 < TP 2 < 2
What is the excluded area?
What can we use for 2 < TP3 < 1?
What is the excluded area?
What about TP4 > 1
What is the excluded area?
Now we can know something of the shape. Note that we really don’t know where the
turn around points are nor what the x at the turn around is. We find those later, in Cal I,
using derivatives.
Note, too, that each factor in the above polynomial was a linear factor: power of 1.
Each factor, then, at the zero, acted like a line…and if we restricted our vision to a tiny
circle right at the intercept it would look like a line, too. Let me sketch in the “zoom”
process I’m talking about…we’ll come back to this idea soon.
4
Suppose we have a polynomial like this one?
f ( x )  ( x  2) 2 ( x  3)
Leading term and end behavior?
Zero’s and y intercept?
Excluded region analysis?
Pick the three test points and let’s look at what’s supposed to happen…it will look odd
until we realize exactly what that “( x  2) 2 ” term is doing to the numbers. “Zoom” in on
it and note that it’s acting – locally only – like a parabola.
5
Now let’s look at one with a cubic term, a quadratic term and a linear term:
f ( x)  x 3 (x  3) 2 (x  6)
Leading term and end behavior?
y intercept?
last term?
constant?
Zeros and behavior of factors?
6
On another note – not explicitly discussed in your book:
The degree can tell you the maximum number of turn arounds that your polynomial
might have…you get a maximum of one fewer than the degree. So if you have a
polynomial of degree 7, you’ll have 6 or fewer turn arounds. If you’re looking at a graph
that has four turnarounds you’ll have to write a polynomial with at least an x 5 in it.
Let’s look at
x2
one
x3
none – all I guaranteed is 2 or less…none is less.
Now let’s do some graphing from the functional expression and then we’ll do it
backwards:
Let’s look at f ( x)  x 2 ( x  3) 3 ( x  4)
Leading term and end behavior:
Zeros and y intercept
Factor analysis
Test points and excluded regions
7
Let’s look at another:
f ( x )  x 4  x 3  12x 2
Factor it:
Leading term and end behavior:
Zeros and y intercept
Factor analysis
Test points and excluded regions
8
Now let’s work backwards:
(7,0)
(4,0)
End behavior? Power, positive or negative leading term?
Zeros and y intercept? Factors?
Powers on factors?
Suggested polynomial:
9
Check turn arounds.
One last one – backwards
(1,0)
(3,0)
(0,3)
10